Calculus Examples

$\int_{0}^{2} 4 t - 1 d t$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{2} 4 t d t + \int_{0}^{2} - 1 d t$

Step 2

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int_{0}^{2} t d t + \int_{0}^{2} - 1 d t$

Step 3

By the Power Rule, the integral of $t$ with respect to $t$ is $\frac{1}{2} t^{2}$ .

$4 (\frac{1}{2} t^{2}]_{0}^{2}) + \int_{0}^{2} - 1 d t$

Step 4

Combine $\frac{1}{2}$ and $t^{2}$ .

$4 (\frac{t^{2}}{2}]_{0}^{2}) + \int_{0}^{2} - 1 d t$

Step 5

Apply the constant rule.

$4 (\frac{t^{2}}{2}]_{0}^{2}) + - t]_{0}^{2}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{t^{2}}{2}$ at $2$ and at $0$ .

$4 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}) + - t]_{0}^{2}$

Step 6.2

Evaluate $- t$ at $2$ and at $0$ .

$4 (\frac{2^{2}}{2} - \frac{0^{2}}{2}) + - 1 \cdot 2 + 0$

Step 6.3

Simplify.

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Step 6.3.1

Raise $2$ to the power of $2$ .

$4 (\frac{4}{2} - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.2

Cancel the common factor of $4$ and $2$ .

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Step 6.3.2.1

Factor $2$ out of $4$ .

$4 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.2.2.2

Cancel the common factor.

$4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.2.2.3

Rewrite the expression.

$4 (\frac{2}{1} - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.2.2.4

Divide $2$ by $1$ .

$4 (2 - \frac{0^{2}}{2}) - 1 \cdot 2 + 0$

Step 6.3.3

Raising $0$ to any positive power yields $0$ .

$4 (2 - \frac{0}{2}) - 1 \cdot 2 + 0$

Step 6.3.4

Cancel the common factor of $0$ and $2$ .

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Step 6.3.4.1

Factor $2$ out of $0$ .

$4 (2 - \frac{2 (0)}{2}) - 1 \cdot 2 + 0$

Step 6.3.4.2

Cancel the common factors.

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Step 6.3.4.2.1

Factor $2$ out of $2$ .

$4 (2 - \frac{2 \cdot 0}{2 \cdot 1}) - 1 \cdot 2 + 0$

Step 6.3.4.2.2

Cancel the common factor.

$4 (2 - \frac{2 \cdot 0}{2 \cdot 1}) - 1 \cdot 2 + 0$

Step 6.3.4.2.3

Rewrite the expression.

$4 (2 - \frac{0}{1}) - 1 \cdot 2 + 0$

Step 6.3.4.2.4

Divide $0$ by $1$ .

$4 (2 - 0) - 1 \cdot 2 + 0$

Step 6.3.5

Multiply $- 1$ by $0$ .

$4 (2 + 0) - 1 \cdot 2 + 0$

Step 6.3.6

Add $2$ and $0$ .

$4 \cdot 2 - 1 \cdot 2 + 0$

Step 6.3.7

Multiply $4$ by $2$ .

$8 - 1 \cdot 2 + 0$

Step 6.3.8

Multiply $- 1$ by $2$ .

$8 - 2 + 0$

Step 6.3.9

Add $- 2$ and $0$ .

$8 - 2$

Step 6.3.10

Subtract $2$ from $8$ .

$6$

Step 7